I really don't think Sol would experience the same upheaval that the rest of the GTVA did when it was cut off. To make a long analysis short: Sol would have existed on its own, by itself, for thousands of years. The rest of the GTVA would have had Sol to rely on for their entire existence (and all communications protocols and military functions and governmental things would assume Sol as the center of society).
Or another example: Say the US was cut in half and the west could no longer communicate with the east. On the east side, the Federal government in Washington D.C. remains in control. All of the states report to it; the military reports to it; all of the central legal and judicial stuff remains intact. There is centralized response to the critical economic and logistic problems caused by splitting the country in half.
On the west side, it's total chaos. Who's the dominant state? California has the largest economy, but Colorado has the military connections. The eastmost states have the most history. Up above the state level, there is no government. Somebody has to take charge, organize, and come up with a system that everybody agrees upon - probably based on the government back in Washington D.C. But that'll take time, and there will be lots of disagreements, and lots of people saying that this thing is biased, and people saying that we need to do things better this time, and so on and so forth. Meanwhile, all kinds of court cases will get decided, laws will get interpreted, inter-state government organizations will form to handle interaction, all with the assumption that they are THE top dog. And it's entirely possible that the states won't be interested in giving up so much power now that they have their independence, and will instead form a loosely-affiliated confederation.
So to summarize, I think Sol would be fairly stable by the time the GTVA reestablished contact.
Still, you're right that they could have better technology. I rely on that assumption because (1) it makes the plan work and (2) I expect that GTVA Command would underestimate the preparation of Earth forces. It's demonstrated overconfidence on many occasions and it's human nature to remember things as they were when you haven't seen them for a long time. I doubt that's what people would actually say, but you'd probably have Command making decisions based on several different reports, and picking the one that seemed the most 'realistic'.
Still, let's play on that. The GTVA opens the portal to earth and finds a technically superior fleet. Assuming that they're still not interested in sharing power, there's a few options they could use.
One is to claim that tactics and strategy have evolved since then, and that the GTVA has superior military intelligence and analysis from Shivan forces. Also that GTVA forces are more battle-experienced. Thus Earth forces could be kept off of the front lines and kept from gaining any glory to rival the GTVA. However, it's pretty weak. The first thing people will say when they see the shiny new Earth ships is, get them to the front lines. Keeping them around when they could obviously be used (and save the lives that people in the GTVA care about, at the cost of lives that people in the GTVA don't even know about) could get some public backlash going, and the Sol forces would be in a much stronger position to force the GTVA into letting them get their way.
Another is to get Sol to sign on as a signatory under the GTVA. In fact, this could already have been done under BETAC, by having some low-ranking government guy fill in for the president "...during a time of crisis, when the president and other reigning government officials are incapacitated or so rendered incapable fulfill any official duties for an indefinite period of time..." and sign the Sol government up for it whether it liked it or not, to add some legitimacy to the document. However, that wouldn't really fly either, if Sol really wanted to come back as the Terran head of the GTVA. Still, the peer pressure would be a pretty powerful tool - the GTVA could paint earth as uncooperative and refuse to deal with it or allow military forces passage through its system. Then start blaming things on Earth if the war started to go badly, and force it to give up portions of its military technology in exchange for cooperation.
The GTVA could stand aside, send the Sol ships to the front, and let them die fighting - whether by actual failure or by some kind of cooked-up "communications failure" due to "unexpected technical difficulties in the nebular environment." But that runs the risk of an outpouring of sympathy for Earth. However, it would hurt Sol's appearance as a leader, somewhat, if its forces rushed in and got themselves killed. It also wouldn't be in a good position to stage any kind of subtle or unsubtle military coup. So it's a mixed bag.
Or the GTVA could stage an incident, have the Sol forces engage in combat with the Vasudan forces, play on fears of a resurgence by the NTF, and blockade and then invade Earth in order to "maintain the peace" (No, this is not meant to be a commentary on current events). Presumably numerical superiority and the Colossus would get the GTVA through this one, or they'd be screwed and would get their asses kicked by Sol, while the Shivans encroach upon the retreating GTVA fleets in the nebula and wipe out the GTVA (and then Sol) from the other direction.
But who knows, maybe I've got the GTVA all wrong and they just wanted Earth to come in and solve all their problems.